Diffuse orthogonally additive operators

Abstract

A regular orthogonally additive operator is a diffuse operator if it is disjoint from all operators in the band generated by the disjointness preserving operators. We present a criterion for principal lateral projections in an order complete vector lattice $E$ to be disjoint. We also state a criterion for a regular orthogonally additive operator to be diffuse. A criterion for the regularity of an integral Urysohn operator acting on ideal spaces of measurable functions is also presented. This criterion is used to show that an integral operator is diffuse. Examples of vector lattices are considered in which the sets of diffuse operators consist only of the zero element. The general form of an order projection operator onto the band generated by the disjointness preserving operators is found. Bibliography: 47 titles.